Coarse sand good. If possible, trench for proper drainage if there is no natural slope. You could toss in a couple of pallets temporarily for them to get up out of the mud. Leaves are free and would work to soak up water. Then they will have a pile of damp leaves to root through. Mine love to root around in wet leaves.

Our climate doesn't usually involve a lot of wet weather. But when we have "one of those years" I keep bags of leaves on hand (ask everyone you know who rakes them up to save them, can always be used in compost pile if you don't end up using them in the run) to throw down after a storm, lots of fresh shavings in the nest boxes to prevent dirty eggs, and as a last resort I'll buy a bale of bedding straw as needed to spread in the run. Sometimes you can score some free bedding straw after Halloween or Christmas at local schools and churches who host holiday celebrations.

I also had a muddy run so i headed to my local yard supply store and bought about 8 bags of pine bark and laid that down. It absorbs the water and there's virtually no mud just have to rake it around when they scratch around in it.